But I see this all the time.

"Considering that correlational analyses indicated high correlations between two pairs of variables: the MPAI Emotion scale and PCL-22, as well as the Cognistat Memory scale and CIQ Productivity factor, the first run of logistic regression analyses included seven variables and two interactions as predictors of employment status. Given that interactions did not significantly predict employment, logistic regression was rerun without the interactions"

In this case they are interactions. But they are still dropping variables. My point is, if you should not drop variables why is the practice so common in the literature.